I agree with WallyDW that a Raspberry Pi Zero W is a bit underpowered even as a satellite (where it is listening for keyword but then passing the serious processing off to a base station). It does do the job, but a Raspberry Pi 3A+ or the new RasPi Zero 2 W respond noticeably quicker to your wakeword, so you don’t have to pause so long before giving your command, and without needing the extra I/O hardware and expense of a RasPi 3B.
I suspect that (while several devices are mentioned in the Rhasspy documentation) there is no formal recommendation for satellite hardware because no combination has yet proved significantly better
I personally have a reSpeaker 4-mic HAT, reSpeaker 2-mic HAT, adafruit Voice Bonnet … and they are all made from almost identical hardware and use the same seeed driver.
While these multi-mic devices have the hardware capability, it appears to be left to system integrators to add Digital Signal Processing (DSP) software into their products. It is the DSP software which will integrate the multiple mics and provide Acoustic Echo Cancellation (AEC), Beamforming, Noise Suppression (NS), etc. As Rolyan has often pointed out, we are tricking ourselves by assuming that multiple mics are automatically better; when it is actually the DSP software which is key. Without DSP, we might as well use a USB sound card with regular microphone and speaker - and yes, one of my satellites is using that and giving much the same result.
I am hoping that the ESP32-S3 (discussed in this thread) will prove to have the desired AI and DSP features - and at a much better price than a RasPi with multi-mic HAT.
As for the base station, I have only a 2 bedroom apartment with 2 humans, and I am still happily using a RasPi 4 running Home Assistant OS and the Rhasspy add-on …though I expect that upgrading to a “better” PC is in my future